Calculus Examples

$x = 2 \ln (y^{2} - 3)$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (x) = \frac{d}{d x} (2 \ln (y^{2} - 3))$

Step 2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$1$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Since $2$ is constant with respect to $x$ , the derivative of $2 \ln (y^{2} - 3)$ with respect to $x$ is $2 \frac{d}{d x} [\ln (y^{2} - 3)]$ .

$2 \frac{d}{d x} [\ln (y^{2} - 3)]$

Step 3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = y^{2} - 3$ .

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Step 3.2.1

To apply the Chain Rule, set $u_{1}$ as $y^{2} - 3$ .

$2 (\frac{d}{d u_{1}} [\ln (u_{1})] \frac{d}{d x} [y^{2} - 3])$

Step 3.2.2

The derivative of $\ln (u_{1})$ with respect to $u_{1}$ is $\frac{1}{u_{1}}$ .

$2 (\frac{1}{u_{1}} \frac{d}{d x} [y^{2} - 3])$

Step 3.2.3

Replace all occurrences of $u_{1}$ with $y^{2} - 3$ .

$2 (\frac{1}{y^{2} - 3} \frac{d}{d x} [y^{2} - 3])$

Step 3.3

Differentiate using the Sum Rule.

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Step 3.3.1

Combine $\frac{1}{y^{2} - 3}$ and $2$ .

$\frac{2}{y^{2} - 3} \frac{d}{d x} [y^{2} - 3]$

Step 3.3.2

By the Sum Rule, the derivative of $y^{2} - 3$ with respect to $x$ is $\frac{d}{d x} [y^{2}] + \frac{d}{d x} [- 3]$ .

$\frac{2}{y^{2} - 3} (\frac{d}{d x} [y^{2}] + \frac{d}{d x} [- 3])$

Step 3.4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = y$ .

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Step 3.4.1

To apply the Chain Rule, set $u_{2}$ as $y$ .

$\frac{2}{y^{2} - 3} (\frac{d}{d u_{2}} [{u_{2}}^{2}] \frac{d}{d x} [y] + \frac{d}{d x} [- 3])$

Step 3.4.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 2$ .

$\frac{2}{y^{2} - 3} (2 u_{2} \frac{d}{d x} [y] + \frac{d}{d x} [- 3])$

Step 3.4.3

Replace all occurrences of $u_{2}$ with $y$ .

$\frac{2}{y^{2} - 3} (2 y \frac{d}{d x} [y] + \frac{d}{d x} [- 3])$

Step 3.5

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

$\frac{2}{y^{2} - 3} (2 y y' + \frac{d}{d x} [- 3])$

Step 3.6

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$\frac{2}{y^{2} - 3} (2 y y' + 0)$

Step 3.7

Combine fractions.

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Step 3.7.1

Add $2 y y'$ and $0$ .

$\frac{2}{y^{2} - 3} (2 y y')$

Step 3.7.2

Combine $2$ and $\frac{2}{y^{2} - 3}$ .

$\frac{2 \cdot 2}{y^{2} - 3} (y y')$

Step 3.7.3

Multiply $2$ by $2$ .

$\frac{4}{y^{2} - 3} (y y')$

Step 3.7.4

Combine $y$ and $\frac{4}{y^{2} - 3}$ .

$\frac{y \cdot 4}{y^{2} - 3} y'$

Step 3.7.5

Combine $\frac{y \cdot 4}{y^{2} - 3}$ and $y'$ .

$\frac{y \cdot 4 y'}{y^{2} - 3}$

Step 3.7.6

Move $4$ to the left of $y$ .

$\frac{4 y y'}{y^{2} - 3}$

Step 4

Reform the equation by setting the left side equal to the right side.

$1 = \frac{4 y y'}{y^{2} - 3}$

Step 5

Solve for $y'$ .

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Step 5.1

Rewrite the equation as $\frac{4 y y'}{y^{2} - 3} = 1$ .

$\frac{4 y y'}{y^{2} - 3} = 1$

Step 5.2

Multiply both sides by $y^{2} - 3$ .

$\frac{4 y y'}{y^{2} - 3} (y^{2} - 3) = 1 (y^{2} - 3)$

Step 5.3

Simplify.

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Step 5.3.1

Simplify the left side.

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Step 5.3.1.1

Cancel the common factor of $y^{2} - 3$ .

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Step 5.3.1.1.1

Cancel the common factor.

$\frac{4 y y'}{y^{2} - 3} (y^{2} - 3) = 1 (y^{2} - 3)$

Step 5.3.1.1.2

Rewrite the expression.

$4 y y' = 1 (y^{2} - 3)$

Step 5.3.2

Simplify the right side.

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Step 5.3.2.1

Multiply $y^{2} - 3$ by $1$ .

$4 y y' = y^{2} - 3$

Step 5.4

Divide each term in $4 y y' = y^{2} - 3$ by $4 y$ and simplify.

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Step 5.4.1

Divide each term in $4 y y' = y^{2} - 3$ by $4 y$ .

$\frac{4 y y'}{4 y} = \frac{y^{2}}{4 y} + \frac{- 3}{4 y}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $4$ .

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Step 5.4.2.1.1

Cancel the common factor.

$\frac{4 y y'}{4 y} = \frac{y^{2}}{4 y} + \frac{- 3}{4 y}$

Step 5.4.2.1.2

Rewrite the expression.

$\frac{y y'}{y} = \frac{y^{2}}{4 y} + \frac{- 3}{4 y}$

Step 5.4.2.2

Cancel the common factor of $y$ .

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Step 5.4.2.2.1

Cancel the common factor.

$\frac{y y'}{y} = \frac{y^{2}}{4 y} + \frac{- 3}{4 y}$

Step 5.4.2.2.2

Divide $y'$ by $1$ .

$y' = \frac{y^{2}}{4 y} + \frac{- 3}{4 y}$

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Simplify each term.

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Step 5.4.3.1.1

Cancel the common factor of $y^{2}$ and $y$ .

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Step 5.4.3.1.1.1

Factor $y$ out of $y^{2}$ .

$y' = \frac{y \cdot y}{4 y} + \frac{- 3}{4 y}$

Step 5.4.3.1.1.2

Cancel the common factors.

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Step 5.4.3.1.1.2.1

Factor $y$ out of $4 y$ .

$y' = \frac{y \cdot y}{y \cdot 4} + \frac{- 3}{4 y}$

Step 5.4.3.1.1.2.2

Cancel the common factor.

$y' = \frac{y \cdot y}{y \cdot 4} + \frac{- 3}{4 y}$

Step 5.4.3.1.1.2.3

Rewrite the expression.

$y' = \frac{y}{4} + \frac{- 3}{4 y}$

Step 5.4.3.1.2

Move the negative in front of the fraction.

$y' = \frac{y}{4} - \frac{3}{4 y}$

Step 6

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = \frac{y}{4} - \frac{3}{4 y}$